Research Article A Study of Cho-Kwon-Srivastava Operator with Applications to Generalized Hypergeometric Functions

Transcription

1 International Mathematics and Mathematical Sciences, Article ID , 6 pages Research Article A Study of Cho-Kwon-Srivastava Operator with Applications to Generalized Hypergeometric Functions F. Ghanim 1 and M. Darus 2 1 Department of Mathematics, College of Sciences, University of Sharjah, Sharjah, UAE 2 School of Mathematical Sciences, Faculty of Science and Technology, Universiti Kebangsaan Malaysia, 43600Bangi,Selangor,Malaysia Correspondence should be addressed to M. Darus; maslina@ukm.edu.my Received 16 May 2014; Accepted 21 June 2014; Published 9 July 2014 AcademicEditor:HariM.Srivastava Copyright 2014 F. Ghanim and M. Darus. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. We introduce a new class of meromorphically analytic functions, which is defined by means of a Hadamard product or convolution) involving some suitably normalized meromorphically functions related to Cho-Kwon-Srivastava operator. A characterization property giving the coefficient bounds is obtained for this class of functions. The other related properties, which are investigated in this paper, include distortion and the radii of starlikeness and convexity. We also consider several applications of our main results to generalized hypergeometric functions. 1. Introduction A meromorphic function is a single-valued function that is analytic in all but possibly a discrete subset of its domain, and at those singularities it must go to infinity like a polynomial i.e., these exceptional points must be poles and not essential singularities). A simpler definition states that a meromorphic function fz) is a function of the form f z) = g z) h z), 1) where gz) and hz) are entire functions with hz) =0 see [1, page 64]). A meromorphic function therefore may only have finite-order, isolated poles and zeros and no essential singularities in its domain. An equivalent definition of a meromorphic function is a complex analytic map to the Riemann sphere. For example, the gamma function is meromorphicinthewholecomplexplanec. In the present paper, we initiate the study of functions which are meromorphic in the punctured disk U ={z:0< z < 1} with a Laurent expansion about the origin; see [2]. Let A be the class of analytic functions hz) with h0) = 1, which are convex and univalent in the open unit disk U = U {0}and for which R {h z)} >0, z U ). 2) For functions f and g analytic in U, wesaythatf is subordinate to g and write f gin U or f z) gz), z U ) 3) if there exists an analytic function wz) in U such that w z) z, fz) =gw z)), z U ). 4) Furthermore, if the function g is univalent in U,then f z) gz) f 0) =g0), f U) gu), z U ). This paper is divided into two sections; the first introduces a new class of meromorphically analytic functions, which is defined by means of a Hadamard product or 5)

2 2 International Mathematics and Mathematical Sciences convolution) involving linear operator. The second section highlights some applications of the main results involving generalized hypergeometric functions. 2. Preliminaries Let Σ denote the class of meromorphic functions fz) normalized by f z) = 1 z + a n z n, 6) which are analytic in the punctured unit disk U ={z:0< z < 1}. For0 β, we denote by S β) and kβ) the subclasses of Σ consisting of all meromorphic functions which are, respectively, starlike of order β andconvexoforderβ in U. For functions f j z) j = 1; 2) defined by f j z) = 1 z + a n,j z n, 7) we denote the Hadamard product or convolution) of f 1 z) and f 2 z) by f 1 f 2 )= 1 z + a n,1 a n,2 z n. 8) Cho et al. [3] and Ghanim and Darus [4]studiedthefollowing function: q, z) = 1 n+1+ ) z n, > 0, 0). 9) Corresponding to the function q, z) and using the Hadamard product for fz) Σ, we define a new linear operator L, ) on Σ by L, f z) =fz) q, z)) = 1 n++1 ) a n zn. 10) The Hadamard product or convolution of the functions f given by 10)withthefunctionsL t,a g and L t,a h given, respectively, by L, g z) = 1 n++1 ) b n zn, z U, gz) Σ), L, h z) = 1 n++1 ) c n zn, z U, hz) Σ), 11) can be expressed as follows: L, f g) z) = 1 n++1 ) a nb n zn, z U ), L, f h) z) = 1 n++1 ) a nc n zn, z U ). 12) By applying the subordination definition, we introduce here a new class Σ ρ,a,b)ofmeromorphic functions, which is defined as follows. Definition 1. Afunctionf Σof the form 6)issaidtobein the class Σ ρ,a,b)if it satisfies the following subordination property: ρ L, f g) z) z ρ A B) L, f h) z) 1+Bz, z U ), 13) where 1 B<A 1, ρ>0, with condition 0 c n b n and L, )f h)z) =0. As for the second result of this paper on applications involving generalized hypergeometric functions, we need to utilize the well-known Gaussian hypergeometric function. One denotes φα, β; z) the class of the function given by φα,β;z)= 1 z + α) n+1 z n, 14) β) n+1 for β =0, 1, 2,...,andα C\{0},where)n = + 1) n+1 is the Pochhammer symbol. We note that where n=0 φα, β; z) = 1 z 2F 1 1, α, β; z), 15) 2 F 1 b,α,β;z)= n=0 b) n α) n β) n z n n! 16) isthewell-knowngaussianhypergeometricfunction. Corresponding to the functions φα, β; z) and q, z) givenin9) andusingthehadamardproductforfz) Σ, we define a new linear operator Lα, β,, ) on Σ by Lα,β,,)fz) =fz) φα,β;z) q, z)) = 1 z + α) n+1 β) n+1 n++1 ) a n zn. 17) The meromorphic functions with the generalized hypergeometric functions were considered recently by Cho and Kim [5], Dziok and Srivastava [6, 7], Ghanim [8], Ghanim et al. [9, 10], and Liu and Srivastava [11, 12].

3 International Mathematics and Mathematical Sciences 3 Now, it follows from 17)that zlα,β,,)fz)) = αl α + 1, β,, ) f z) α+1) L α, β,, ) f z). 18) ρbb n +c n {A B) ρb})a n z n+1 ) 1 ρ a n++1 ) n b n c n ) zn+1 The subordination relation 13) in conjunction with 17) takes the following form: ρ Lα+1,β,,)fz) Lα,β,,)fz) ρ A B) z 1+Bz, 0 B<A 1, ρ>0). 19) Definition 2. Afunctionf Σof the form 6) issaidtobe in the class Σ ρ,α,β,a,b) if it satisfies the subordination relation 19)above. 1. A B) n++1 ) 1 ρbb n +c n {A B) ρb})a n zn+1 ) 22) 3. Characterization and Other Related Properties In this section, we begin by proving a characterization property which provides a necessary and sufficient condition for a function f Σof the form 6) tobelongtotheclass Σ ρ,a,b)of meromorphically analytic functions. Theorem 3. The function f Σissaid to be a member of the class Σ ρ, A, B) if and only if it satisfies n++1 ) ρ b n 1+B) c n ρ 1+B) + A B)) a n A B. 20) The equality is attained for the function f n z) given by f n z) = 1 z A B)n++1) + ρ b n 1+B) c n ρ 1+B) + A. B))zn 21) Proof. Let f of the form 6) belongtotheclassσ ρ,a,b). Then, in view of 12), we find that ρ n++1 ) a n b n c n )z n+1 A B) n++1 ) Putting z = r 0 r < 1) and noting the fact that the denominator in the above inequality remains positive by virtue of the constraints stated in 13) forallr [0,1),we easily arrive at the desired inequality 20)bylettingz 1. Conversely, if we assume that the inequality 20) holds true in the simplified form 22), it can readily be shown that ρ{f g)z)) f h) z))} <1, ρb f g) z))+{ρa B) ρb} f h) z)) z U ), 23) whichisequivalenttoourconditionoftheorem,sothatf Σ ρ,a,b), hence the theorem. Theorem 3 immediately yields the following result. Corollary 4. If the function f Σ belongs to the class Σ ρ,a,b),then a n A B)n++1) ρ b n 1+B) c n 1, n ρ 1+B) +A B)), 24) where the equality holds true for the functions f n z) given by 21). We now state the following growth and distortion properties for the class Σ ρ, A, B). Theorem 5. If the function f defined by 6) is in the class Σ ρ,a,b),then,for0< z =r<1,onehas 1 r A B)2+) ρ b 1 1+B) c 1 ρ 1+B) +A B))r f z) 1 r + A B)2+) ρ b 1 1+B) c 1 ρ 1+B) +A B))r,

4 4 International Mathematics and Mathematical Sciences 1 r 2 A B)2+) ρ b 1 1+B) c 1 ρ 1+B) + A B)) f z) 1 r 2 + A B)2+) ρ b 1 1+B) c 1 ρ 1+B) +A B)). 25) Proof. Since f Σ ρ,a,b), Theorem 3 readily yields the inequality a n A B)2+) ρ b 1 1+B) c 1 ρ 1+B) +A B)). 26) Thus, for 0< z =r<1and utilizing 26), we have f z) = 1 z + a n z n 1 r +r a n 1 r +r A B)2+) ρ b 1 1+B) c 1 ρ 1+B) + A B)), f z) = 1 z a n z n 1 r r a n 1 r r A B)2+) ρ b 1 1+B) c 1 ρ 1+B) + A B)). 27) Also from Theorem 3,we get Hence n a n A B)2+) ρ b 1 1+B) c 1 f z) = 1 z 2 + n a n z n 1 1 r 2 + a n ρ 1+B) +A B)). 28) 1 r 2 + A B)2+) ρ b 1 1+B) c 1 ρ 1+B) +A B)), f z) = 1 z 2 n a n z n 1 1 r 2 n a n 1 r 2 A B)2+) ρ b 1 1+B) c 1 ρ 1+B) +A B)). 29) This completes the proof of Theorem 5. We next determine the radii of meromorphic starlikeness and meromorphic convexity of the class Σ ρ, A, B), which are given by Theorems 6 and 7 below. Theorem 6. If the function f defined by 6) is in the class Σ ρ,a,b),thenf is meromorphic starlike of order δ in the disk z < r 1,where r 1 = inf n 1 {1 δ) ρ b n 1+B) c n ρ 1+B) +A B)) A B)n+2 δ)) 1 } 1/n+1). The equality is attained for the function f n z) given by 21). Proof. It suffices to prove that zfz)) f z) For z < r 1,wehave 30) +1 1 δ. 31) zfz)) +1 f z) = n+1)/n )) a n z n 1/z + /n )) a n z n = n+1)/n + + 1)) a n z n+1 1+ /n + + 1)) a n z n+1 n+1)/n + + 1)) a n z n+1 1+ /n + + 1)) a n z n+1. Hence 32)holdstruefor or n+1) n++1 ) a n z n+1 1 δ)1 n++1 ) a n z n+1 ) 32) 33) n+2 δ)/n + + 1)) a n z n ) 1 δ) With the aid of 20)and34), it is true to say that for fixed n n+2 δ)/n )) z n+1 1 δ) n++1 ) ρ b n 1+B) + c n ρ 1+B) +A B)) A B) 1, n 1. 35)

5 International Mathematics and Mathematical Sciences 5 Solving 35)for z,weobtain z {1 δ) ρ b n 1+B) + c n ρ 1+B) + A B)) n+2 δ)a B)) 1 } n+1. 36) The following consequences of Theorem 8 can be deduced by applying 39)and40)alongwithDefinition2. Corollary 9. If the function f Σ belongs to the class Σ ρ,α,β,a,b),then a n This completes the proof of Theorem 6. Theorem 7. If the function f defined by 6) is in the class Σ ρ,a,b),thenf is meromorphic convex of order δ in the disk z < r 2,where r 2 = inf n 1 {1 δ) ρ b n 1+B) c n ρ 1+B) + A B)) n n+2 δ)a B)) 1 } 1/n+1). The equality is attained for the function f n z) given by 21). 37) Proof. Byusingthesametechniqueemployedintheproofof Theorem 6,we can show that zf z)) f z)) 1 δ. 38) +2 For z < r 1 and with the aid of Theorem 3, wehavethe assertion of Theorem Applications Involving Generalized Hypergeometric Functions Theorem 8. The function f Σissaid to be a member of the class Σ ρ,α,β,a,b)if and only if it satisfies ρ b n 1+B) c n ρ 1+B) + A B)) α) n+1 β) n+1 n++1 ) a n A B. The equality is attained for the function f n z) given by f n z) = 1 z A B)n++1) + ρ b n 1+B) c n ρ 1+B) + A, B))zn 39) n 1. 40) Proof. Byusingthesametechniqueemployedintheproofof Theorem 3 along with Definition 2,wecanproveTheorem 8. A B)n++1) β) n+1 ρ b n 1+B) c n ρ 1+B) +A B))α), n+1 n 1, 41) where the equality holds true for the functions f n z) given by 40). Corollary 10. If the function f defined by 6) is in the class Σ ρ,α,β,a,b),thenf is meromorphic starlike of order δ in the disk z < r 3,where r 3 = inf n 1 {1 δ) ρ b n 1+B) c n ρ 1+B) +A B)) A B)n+2 δ)) 1 } 1/n+1). The equality is attained for the function f n z) given by 40). 42) Corollary 11. If the function f defined by 6) is in the class Σ ρ,α,β,a,b),thenf is meromorphic convex of order δ in the disk z < r 4,where r 4 = inf n 1 {1 δ) ρ b n 1+B) c n ρ 1+B) + A B)) n n+2 δ)a B)) 1 } 1/n+1). The equality is attained for the function f n z) given by 40). Conflict of Interests 43) The authors declare that there is no conflict of interests regarding the publication of this paper. Authors Contribution All authors read and approved the final paper. Acknowledgment The work here was fully supported by FRGSTOPDOWN/ 2013/ST06/UKM/01/1. References [1] S. G. Krantz, Meromorphic functions and singularities at infinity, in Handbook of Complex Variables, pp.63 68, Birkhäauser, Boston, Mass, USA, 1999.

6 6 International Mathematics and Mathematical Sciences [2] A. W. Goodman, Functions typically-real and meromorphic in the unit circle, Transactions of the American Mathematical Society,vol.81,pp ,1956. [3]N.E.Cho,O.S.Kwon,andH.M.Srivastava, Inclusionand argument properties for certain subclasses of meromorphic functions associated with a family of multiplier transformations, Mathematical Analysis and Applications, vol. 300, no. 2, pp , [4] F. Ghanim and M. Darus, Some properties on a certain class of meromorphic functions related to Cho-Kwon-Srivastava operator, Asian-European Mathematics,vol.5,no.4, Article ID , pp. 1 9, [5]N.E.ChoandI.H.Kim, Inclusionpropertiesofcertain classes of meromorphic functions associated with the generalized hypergeometric function, Applied Mathematics and Computation,vol.187,no.1,pp ,2007. [6]J.DziokandH.M.Srivastava, Somesubclassesofanalytic functions with fixed argument of coefficients associated with the generalized hypergeometric function, Advanced Studies in Contemporary Mathematics Kyungshang),vol.5,no.2,pp , [7] J. Dziok and H. M. Srivastava, Certain subclasses of analytic functions associated with the generalized hypergeometric function, Integral Transforms and Special Functions,vol.14,no.1,pp. 7 18, [8] F. Ghanim, A study of a certain subclass of Hurwitz-Lerch- Zeta function related to a linear operator, Abstract and Applied Analysis,vol.2013,ArticleID763756,7pages,2013. [9] F. Ghanim and M. Darus, A new class of meromorphically analytic functions with applications to the generalized hypergeometric functions, Abstract and Applied Analysis, vol. 2011, Article ID , 10 pages, [10] F. Ghanim, M. Darus, and Z.-G. Wang, Some properties of certain subclasses of meromorphically functions related to chokwon-srivastava operator, Information Journal, vol. 16, no. 9, pp , [11] J. Liu and H. M. Srivastava, A linear operator and associated families of meromorphically multivalent functions, Mathematical Analysis and Applications,vol.259,no.2,pp , [12] J. Liu and H. M. Srivastava, Classes of meromorphically multivalent functions associated with the generalized hypergeometric function, Mathematical and Computer Modelling, vol. 39, no. 1,pp.21 34,2004.

7 Advances in Operations Research Advances in Decision Sciences Applied Mathematics Algebra Probability and Statistics The Scientific World Journal International Differential Equations Submit your manuscripts at International Advances in Combinatorics Mathematical Physics Complex Analysis International Mathematics and Mathematical Sciences Mathematical Problems in Engineering Mathematics Discrete Mathematics Discrete Dynamics in Nature and Society Function Spaces Abstract and Applied Analysis International Stochastic Analysis Optimization